GMAT number properties is the single most-tested arithmetic concept on the Quantitative Reasoning section, accounting for roughly 13.83% of Quant questions. This guide walks through every rule that actually shows up on test day—integers, primes, divisibility, odds/evens, factors, LCM/GCD, and remainders—and pairs each one with the trap GMAT writers use to catch you out.
The GMAT Focus Edition Quantitative Reasoning section runs 21 questions in 45 minutes, all in Problem Solving format. Number properties—often labeled "GMAT integer properties" in prep materials—is the most heavily tested arithmetic subtopic. Test-takers should expect roughly 5 to 7 number properties and number theory questions per Quant section, which means a single weak spot here can quietly drag down a whole score band.
Every number properties question on the GMAT falls into one of four buckets: divisibility and primes; odd/even rules; positive, negative, and zero behavior; and factors, multiples, and remainders. Memorize the buckets and you'll recognize the question type within five seconds—which is often the difference between solving cleanly and getting trapped in algebra.
| Topic | Approx. % of Quant Questions | Notes |
|---|---|---|
| Properties of Integers (Number Properties) | ~13.83% | Most-tested single arithmetic subtopic |
| Percents (calculations, % change) | ~8.72% | High frequency, low difficulty |
| Descriptive Statistics (mean, median, mode, range) | ~8.30% | Steady appearance, moderate difficulty |
| Algebra (linear & quadratic equations) | ~15.96% | Second-largest content bucket |
| Arithmetic (overall) | ~38.94% | Includes number properties, percents, and stats |
| Geometry | ~9.15% | Lowest emphasis on the Focus Edition |
Number properties questions reward fluency with definitions and punish assumptions. Test Ninjas notes that these problems are often written in "GMAT code"—you must translate "is n an integer?" into "does the divisibility close cleanly?" before you can solve. The students who struggle here are not the ones who never learned the rules; they're the ones who never trained themselves to slow down at the translation step.
The term integer includes positive whole numbers, negative whole numbers, and zero. Fractions, decimals, π, square roots that don't resolve cleanly, and other non-integer reals are not integers. The GMAT uses the word "integer" deliberately—when it appears, treat it as a hard constraint; when it's absent, treat that absence as equally important.
A prime is a natural number greater than 1 with exactly two distinct positive divisors: 1 and itself. The first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Notice 1 is not prime (only one divisor), and 2 is the only even prime—every other even number has 2 as a non-trivial factor. For two-digit numbers, divisibility by 2, 3, 5, or 7 is enough to determine whether the number is prime.
GMAT divisibility rules are the highest-leverage memorization in the entire Quant section. You'll use them daily—on prime checks, factor counts, LCM/GCD problems, and digit-pattern questions. The nine rules below are the complete set worth knowing cold.
| Divisor | Rule | Quick Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 | 138 → ends in 8 → divisible by 2 |
| 3 | Sum of digits is divisible by 3 | 417 → 4+1+7 = 12 → divisible by 3 |
| 4 | Last two digits form a number divisible by 4 | 5,316 → '16' is divisible by 4 |
| 5 | Last digit is 0 or 5 | 1,245 → ends in 5 → divisible by 5 |
| 6 | Divisible by both 2 and 3 | 234 → even AND digit sum 9 → divisible by 6 |
| 8 | Last three digits form a number divisible by 8 | 9,128 → '128' = 8 × 16 → divisible by 8 |
| 9 | Sum of digits is divisible by 9 | 513 → 5+1+3 = 9 → divisible by 9 |
| 10 | Last digit is 0 | 4,720 → ends in 0 → divisible by 10 |
| 11 | Alternating digit sum is divisible by 11 | 2,728 → (2+2) − (7+8) = −11 → divisible by 11 |
Worked Example
Setup: Is 4,182 divisible by 6?
Pick a divisor from 2 through 11 to see the rule, a quick example, and how often it's relevant on the GMAT.
Odd and even numbers apply only to integers. The arithmetic is mechanical: odd plus odd is even; even plus even is even; odd plus even is odd. For multiplication, any even factor makes the product even, and the only way to get an odd product is to multiply odd by odd. Drilling these GMAT odd and even numbers rules pays off most on Data Sufficiency, where you'll often need to deduce parity from indirect statements.
Zero is an integer. Zero is even (it's divisible by 2). Zero is neither positive nor negative. Forgetting any of these is a top-three reason students get number properties questions wrong, especially when they're picking numbers to test a Data Sufficiency statement and quietly skip 0. Make zero the first value you test, every single time.
Negative times negative is positive. Negative times positive is negative. The same rules hold for division. The traps usually appear inside inequalities: multiplying or dividing both sides by a negative flips the inequality sign. If a variable could be negative, your inequality work is no longer mechanical—you need cases.
| Operation | Result | Why it matters |
|---|---|---|
| odd + odd | even | Two odd unknowns must sum to an even number |
| even + even | even | Useful for proving evenness in DS |
| odd + even | odd | Mixed parity always gives odd |
| odd × odd | odd | Only way to keep oddness through multiplication |
| even × any integer | even | Single even factor forces an even product |
| negative × negative | positive | Sign cancels out—watch for in inequalities |
| negative × positive | negative | Direction flips when used in inequalities |
| zero × any number | zero | Zero is the great equalizer—often the trap value |
Worked Example
Setup: If x and y are integers and xy is odd, what can you conclude about x and y?
A factor divides into a number; a multiple is what you get when you multiply by an integer. So 4 is a factor of 12 (because 4 divides 12 evenly), and 12 is a multiple of 4 (because 12 = 4 × 3). The wording is interchangeable in math but never in GMAT prompts—test writers swap "factor" and "multiple" deliberately to trip up students who skim. Read these words like a contract.
Prime factorization is the universal tool for GMAT LCM and GCD problems. To find the greatest common divisor, take the lowest power of each prime that appears in both numbers. To find the least common multiple, take the highest power of every prime that appears in either number. This recipe never fails, and it scales to three or more numbers without modification.
To count the factors of a positive integer, find its prime factorization, add 1 to each exponent, and multiply. For example, 12 = 2² × 3¹ → (2+1)(1+1) = 6 factors. A useful corollary: perfect squares always have an odd number of distinct factors, because every exponent in their prime factorization is even, so each (exponent + 1) is odd, and odd × odd × … = odd.
Worked Example
Setup: Find the GCD and LCM of 12 and 18, then verify the product identity.
Enter any positive integer up to 10,000 and instantly see its prime factorization, total factor count, and whether it is prime or a perfect square.
When a positive integer a is divided by a positive integer b, the result is a = bq + r, where q is the quotient and r is the remainder, with 0 ≤ r < b. That last constraint is non-negotiable—the remainder is always less than the divisor and never negative on the GMAT. Most GMAT remainders questions reduce to recognizing this identity and substituting cleverly.
Remainder questions on Data Sufficiency are tailor-made for picking numbers. Choose values that satisfy the constraints (e.g., "n leaves remainder 3 when divided by 7" → try n = 3, 10, 17, 24) and check whether the answer is consistent. Algebraic approaches almost always work too, but they are slower and more error-prone, especially under time pressure.
When a problem asks for the remainder of a large power divided by 10 (or for the units digit of a huge expression), use last-digit cycles. The last digits of powers of any single digit cycle in patterns of length 1, 2, or 4. Identify the cycle, find the position of your exponent within it, and read off the answer.
Worked Example
Setup: What is the remainder when 7^85 is divided by 10?
The mistakes below repeat in nearly every prep-class debrief. None require new theory to fix—they require a habit of pausing on the words that look "obvious." Build the pause into your routine and your accuracy on number properties questions climbs immediately.
Students see a variable like x or n and unconsciously imagine 1, 2, 3, 4, 5. GMAT writers exploit this every chance they get. Unless the problem says "x is a positive integer," x can be zero, negative, a fraction, or a decimal. Train yourself to scan for the word "integer" and the word "positive" before you touch the algebra.
"n is a multiple of 6" and "6 is a factor of n" mean the same thing—but students who only half-translate the sentence will sometimes solve for the wrong relationship. When you read a number-properties prompt, rewrite it in the simplest direction (a divides into b) before doing anything else.
Test Ninjas emphasizes that GMAT number properties questions are written in "GMAT code" rather than plain English. "Is n divisible by 12?" really means "does 12 divide into n with remainder 0?" "Is n/6 an integer?" really means "is n a multiple of 6?" These reframings are the actual work of the question—solving comes after.
| Common Mistake | Why it's wrong | Correct Approach |
|---|---|---|
| Assuming x is a positive integer | GMAT rarely says 'integer' unless it matters | Always test 0, negatives, and fractions when not restricted |
| Forgetting that zero is even | Zero satisfies divisibility by 2 and is an integer | Include 0 in odd/even and sign-related test cases |
| Confusing factor and multiple | A factor divides INTO; a multiple comes FROM | Translate the sentence into 'a divides into b' before solving |
| Saying 4x · 6y is always a multiple of 24 | x and y can share factors that overlap with 4 and 6 | Use prime factorization to count distinct primes |
| Skipping the GMAT-code translation | DS questions hide divisibility behind 'is n an integer?' | Rephrase the question as 'does b divide a evenly?' |
Memorize the divisibility rules, the first 10 primes, and the odd/even/sign rules until you can recite them without thinking. Speed matters here because you'll apply them dozens of times per practice section, and any second you spend "looking up" a rule is a second you can't spend on the actual problem.
Note that the GMAT Focus Edition Quantitative Reasoning section is now Problem Solving only, with Data Sufficiency moved to the Data Insights section. But number properties traps appear in both sections, so train across both formats from day one. Practicing 5 to 7 mixed questions per session mirrors the realistic density you'll face and builds the right pacing instincts.
The single highest-leverage habit in GMAT number properties prep is logging which trap you fell into, not just which answer you missed. Categories like "forgot zero," "assumed positive integer," "confused factor/multiple," and "missed the GMAT-code translation" recur. After two weeks, you'll see your top three traps, and you can drill those specifically.
Test prep providers estimate roughly 5 to 7 number properties and number theory questions in the GMAT Quantitative Reasoning section. That makes properties of integers the most heavily tested arithmetic concept, accounting for about 13.83% of Quant questions according to published topic-frequency analyses. If you're trying to budget your study time, this is the highest-yield single subtopic in the entire Quant section.
Yes. On the GMAT, zero is an integer and zero is even. Zero is the only integer that is neither positive nor negative. Forgetting zero is one of the most common mistakes test-takers make on Data Sufficiency questions, especially when picking numbers to test a statement. Always include zero in your test cases unless the problem explicitly excludes it.
Memorize the rules for 2, 3, 4, 5, 6, 8, 9, 10, and 11. The most useful are: divisible by 2 if it ends in an even digit; divisible by 3 if the digit sum is divisible by 3; divisible by 5 if it ends in 0 or 5; divisible by 9 if the digit sum is divisible by 9. For two-digit numbers, you only need to test divisibility by 2, 3, 5, and 7.
The GCD (greatest common divisor) is the largest number that divides into two integers without a remainder; find it by taking the lowest power of each prime they share. The LCM (least common multiple) is the smallest positive integer both numbers divide into; find it by taking the highest power of every prime that appears in either factorization. The product of two numbers always equals their GCD times their LCM.
Remainders appear less often than divisibility and prime factorization, but they show up disproportionately on harder, 700+ level questions. Master the basic identity a = bq + r where 0 ≤ r < b, learn to pick smart numbers, and recognize cyclical patterns in last digits of large powers. Even one remainder question answered correctly can lift your section score notably.
Assuming variables are positive integers when the problem doesn't say so. GMAT writers exploit this by leaving the door open for zero, negatives, fractions, and decimals. The fix is mechanical: every time you see a variable, ask "could this be zero, negative, or non-integer?" If yes, test those cases before committing to an answer, especially on Data Sufficiency.